Problem: Factor the following expression: $-3$ $x^2$ $-7$ $x+$ $40$
Solution: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-3)}{(40)} &=& -120 \\ {a} + {b} &=& & & {-7} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-120$ and add them together. Remember, since $-120$ is negative, one of the factors must be negative. The factors that add up to ${-7}$ will be your ${a}$ and ${b}$ When ${a}$ is ${8}$ and ${b}$ is ${-15}$ $ \begin{eqnarray} {ab} &=& ({8})({-15}) &=& -120 \\ {a} + {b} &=& {8} + {-15} &=& -7 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-3}x^2 +{8}x {-15}x +{40} $ Group the terms so that there is a common factor in each group: $ ({-3}x^2 +{8}x) + ({-15}x +{40}) $ Factor out the common factors: $ x(-3x + 8) + 5(-3x + 8) $ Notice how $(-3x + 8)$ has become a common factor. Factor this out to find the answer. $(-3x + 8)(x + 5)$